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Properties

Label 2-1400-56.45-c0-0-1
Degree $2$
Conductor $1400$
Sign $0.605 + 0.795i$
Analytic cond. $0.698691$
Root an. cond. $0.835877$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (−0.5 − 0.866i)7-s + 0.999·8-s + (0.5 + 0.866i)9-s + (−0.499 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (1.5 + 0.866i)17-s + (0.499 − 0.866i)18-s + (−0.5 − 0.866i)23-s + 0.999·28-s + (1.5 + 0.866i)31-s + (−0.499 + 0.866i)32-s − 1.73i·34-s − 0.999·36-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (−0.5 − 0.866i)7-s + 0.999·8-s + (0.5 + 0.866i)9-s + (−0.499 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (1.5 + 0.866i)17-s + (0.499 − 0.866i)18-s + (−0.5 − 0.866i)23-s + 0.999·28-s + (1.5 + 0.866i)31-s + (−0.499 + 0.866i)32-s − 1.73i·34-s − 0.999·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.605 + 0.795i$
Analytic conductor: \(0.698691\)
Root analytic conductor: \(0.835877\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :0),\ 0.605 + 0.795i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8269939557\)
\(L(\frac12)\) \(\approx\) \(0.8269939557\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 \)
7 \( 1 + (0.5 + 0.866i)T \)
good3 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + 1.73iT - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T + T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 - 1.73iT - T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.03356353414453699103718349205, −8.916920055650879232363472994235, −8.051645803199376206093487782013, −7.50301211291397246961275576037, −6.57608681935525486481620078292, −5.27167970366015021865604844389, −4.23784975096476487722017851375, −3.54415359655160071793357103862, −2.36846604699452952415481155773, −1.11729674022285314441964022612, 1.14555718080571493573882026365, 2.80887601638612593341476237134, 4.00983701707170394747777622196, 5.15349239292167639414577953861, 5.94927884281690563030827749250, 6.54981025084636754142039533373, 7.53180114653832798300357303879, 8.156488497678090914975982233093, 9.228480689166928959682455663129, 9.644215447676611310771336950373

Graph of the $Z$-function along the critical line